Titlebook: Bayesian Compendium; Marcel van Oijen Textbook 2024Latest edition The Editor(s) (if applicable) and The Author(s), under exclusive license

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Deriving the Posterior Distribution,s the information content of our data. So all that is left is to apply Bayes’ Theorem (Eq. (.)) to derive our desired posterior distribution. Note that when talking about the posterior, we use the phrase ‘deriving the’ distribution rather than ‘a(chǎn)ssigning a’ distribution. That is because Bayes’ Theor

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Markov Chain Monte Carlo Sampling (MCMC),or—instead we aim to determine the posterior probability distribution for the parameters. Only the full probability distribution adequately represents our state of knowledge. Although this shift in thinking has made rigorous uncertainty quantification possible, it has also created computational prob

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MCMC and Multivariate Models,fundamentally different from the simpler models we studied in the previous chapters; we can still write them as functions . of their input consisting of covariates . and parameters .. But the output . from the models will be multivariate, e.g. time series of different properties of an ecosystem. Tha

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Discrepancy,ertainty translates into predictive uncertainty. And if we get new data, then we can use Bayes’ Theorem to update the parameter distribution and thereby reduce our predictive uncertainty. So far, so good. But a more difficult problem is that of uncertainty about model structure. We know that all mod

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Gaussian Processes and Model Emulation,MC algorithms. MCMC is especially slow when the model of interest is a process-based model (PBM) with a long run-time. In such cases, it may be good to replace the PBM with a faster surrogate model. The surrogate model will take the same inputs as the original model but calculate the output more qui